Geodesic
Exact formulas in~some boundary crossing problems
Izvestiya. Mathematics , Tome 87 (2023) no. 1, pp. 45-60

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For a wide class of integer-valued random walks, we obtain exact expressions for the distribution of the first excess over level and the corresponding renewal function as well as for the distribution of the trajectory supremum if it is finite. We discuss possibilities of obtaining explicit expressions for pre-stationary and stationary distributions of a random walk with switchings at the strip boundaries. The research is based on the factorization representations for the double moment generating functions of the distributions under study.
Keywords: integer-valued random walk, factorization method, excess over boundary, renewal function, oscillating random walk. 
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     author = {V. I. Lotov},
     title = {Exact formulas in~some boundary crossing problems},
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V. I. Lotov. Exact formulas in~some boundary crossing problems. Izvestiya. Mathematics , Tome 87 (2023) no. 1, pp. 45-60. http://geodesic.mathdoc.fr/item/IM2_2023_87_1_a2/